6 Modeling and Analysis

The analysis of LLR data requires a sophisticated model of the solar system ephemeris that also
includes all the significant effects that contribute to the range between the Earth stations and the
lunar retroreflectors. These models compute a range prediction and the partial derivatives of
range with respect to each model parameter at the epoch of each normal point. The model
predictions take into account orbital parameters, attraction to the Sun and planets, relativistic
corrections, as well as tidal distortions, plate motion, and other phenomena that affect the
position of the retroreflector and ground station relative to the centers of mass of the Earth and
Moon [80]. Some of these parameters are measured by other means, but most are estimated from the
LLR data. The range measurements are corrected for atmospheric delay and a weighted least
square analysis is performed to estimate the 170 parameters in the model [38, 36], most of
which are initial conditions and masses of solar system bodies. LLR data is often combined
with other spacecraft and planetary tracking data to further constrain the estimates or remove
degeneracies.

A number of models have been developed over the past 40 years. The model developed at the Jet
Propulsion Laboratory (JPL) was one of the first programs for LLR analysis [83] and continues to be
updated. It was recently used to produce limits on the SEP violation, time variation of the gravitational
constant, and interior structure of the Moon [80, 81]. The open-source Planetary Ephemeris Program
(PEP) is undergoing a major upgrade for LLR analysis at the Harvard-Smithsonian Center for Astrophysics
(CfA). It was also used for one of the first LLR analyses to test the SEP [63] and was recently used to test
for Lorentz violation using LLR data [5]. A model developed at the University of Hannover was also
recently used to produce limits on the relativity parameters, including the preferred frame PPN
parameters [36, 38].

With the now routine operation of APOLLO at Apache Point, millimeter level data is being
produced [6]. Unfortunately, none of the ephemeris models is currently capable of handling millimeter class
data to maximum advantage [82]. New effects and error reduction techniques that become
important at the millimeter level need to be added to the analysis tools. To take advantage
of the next generation of LLR instruments, these codes will need to be further modified and
rigorous theoretical work will need to be performed to permit tests of new ideas in physics.
Substantial effort is also required to address the multitude of effects that will contribute at that
level.

Many of these effects will be scientifically interesting in their own right. In particular, the lunar interior
models require significant improvement. There are also additional Earth effects that need further model
development, such as the loading of the lithosphere by the atmosphere and ocean, which causes the
observing station to move vertically (and horizontally) with the tides and weather. Models of
these effects are available that are deemed accurate to better than 0.1 mm and tested in VLBI
analysis software at CfA [64], but need to be incorporated into the analysis programs. As ranging
precision is further improved, more sophisticated atmospheric models and auxiliary measurements
will need to be developed. Other important effects for advanced LLR analysis include solar
radiation pressure, thermal cycling of the reflectors, solar tides on the moon, and solar mass
changes [82].